Before the billionaire owners (including from America who are more than any others in it for the money) football in Europe was about the community and still is for many clubs. There is actual competition with relegation and rewards.
American athletes earn a lot more from endorsements compared to their European counterparts; majority of top 10 earners from endorsements are Americans. If they agreed to reduced salary to win, it is because the salary pales in comparison to what they will earn from elsewhere if they are part of a winning project.
Messi and Ronaldo are exceptions mainly due the fact that they are global brands. The pull they have on the audience is unparalleled. Most people around the world don't even know Tom Brady.
Taking all the teams together, European football is richer and its Premier competition, the champions league earns far more than super bowl. However, individual teams are wealthier than most European counterparts because of the closed shop system designed to maximise revenue and a huge focus on advertisement.
Bradys earnings come more from endorsements than from his salary just like most American athletes. He can agree to salary reduction for a winning team because that would mean more endorsements and more money.